Haar null sets and the consistent reflection of non-meagreness

A subset X of a Polish group G is called Haar null if there exists a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g, h ∈ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ(X + t) = 0 for every t ∈ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∈ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalises results of Bartoszyński and Burke-Miller.